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Crystal fields¶
This example explains how to implement crystal fields in edrixs.
We need to import these modules.
import edrixs
import numpy as np
import scipy
np.set_printoptions(precision=2, suppress=True, linewidth=90)
Crystal field matrices¶
Let us start by considering the common case of a \(d\) atomic shell in a
cubic crystal field. This is controlled by parameter \(10D_q\) and is
described in terms of a matrix which we will assign to cfmat. edrixs
can make this matrix via.
Note this matrix is in a complex harmonic basis \(Y^m_l\) where \(m\) goes from \(-l,-l+1,...,l-1, l\). There is an up spin and a down spin for each \(Y^m_l\). This matrix is not diagonal in the complex harmonic basis, but it would be diagonal in the real harmonic basis \(d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}\). Let us diagonalize this matrix as a check and print out the energies and their degeneracies.
e, v = scipy.linalg.eigh(cfmat)
e = e.round(decimals=6)
unique_e = np.unique(e)
degeneracies = [sum(evalue == e) for evalue in unique_e]
print("E \tDegeneracy")
for evalue, degenvalue in zip(unique_e, degeneracies):
print("{:.1f}\t{:.0f}".format(evalue, degenvalue))
print("{} distinct energies".format(len(unique_e)))
E Degeneracy
-4.0 6
6.0 4
2 distinct energies
This makes sense! We see two different energies split by \(10D_q=10\). Let us look at the six columns corresponding to the lower energy eigenvalues.
print(v[:, :6].real)
[[ 0. 0. 0. 0. 0.71 0. ]
[ 0. 0. 0. 0. 0. 0.71]
[ 0. 0. -1. 0. 0. 0. ]
[ 0. 0. 0. -1. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. ]
[ 1. 0. 0. 0. 0. 0. ]
[ 0. 1. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. -0.71 0. ]
[ 0. 0. 0. 0. 0. -0.71]]
These are the set of so-called \(t_{2g}\) orbitals, composed of \(Y^2_2, Y^{-2}_2, Y^{1}_2, Y^{-1}_2\). The rest of the eigenvectors (the last four) are
print(v[:, 6:].real)
[[-0.71 0. 0. 0. ]
[ 0. -0.71 0. 0. ]
[ 0. 0. 0. 0. ]
[ 0. 0. 0. 0. ]
[ 0. 0. 0. 1. ]
[ 0. 0. 1. 0. ]
[ 0. 0. 0. 0. ]
[ 0. 0. 0. 0. ]
[-0.71 0. 0. 0. ]
[ 0. -0.71 0. 0. ]]
These are the set of so-called \(e_{g}\) orbitals, composed of
\(Y^2_2, Y^{-2}_2, Y^{0}_2\). We can use edrixs to prove that
cfmat would be diagonal in the real
harmonic basis. An operator \(\hat{O}\) can be transformed into an
operator in another basis \(\hat{O}^{\prime}\) using a unitary
transformation matrix \(T\) as
\[\hat{O}^{\prime} = (T)^{\dagger} \hat{O} (T).\]
This is computed as follows
cfmat_rhb = edrixs.cb_op(cfmat, edrixs.tmat_c2r('d', ispin=True))
print(cfmat_rhb.real)
[[ 6. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 6. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. -4. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. -4. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. -4. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. -4. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 6. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 6. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. -4. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. -4.]]
where edrixs.tmat_c2r('d', ispin=True) is the transformation matrix.
We needed to tell edrixs that we are working with a \(d\)-shell and that it
should include spin. We could also have transformed v to see how these
eignevectors are composed of the real harmonic basis. We will see an example
of this later.
Crystal field on an atom¶
To simulate the solid state, we need to combine the crystal field with Coulomb interactions. Let us choose an atomic model for Ni.
Let us implement a tetragonal crystal field, for which we need to pass
d1 the splitting of \(d_{yz}/d_{zx}\) and \(d_{xy}\) and
d3 the splitting of \(d_{3z^2-r^2}\) and \(d_{x^2-y^2}\).
To determine the eigenvalues and eigenvectors we need to transform both our Coulomb matrix and our crystal field matrix into the same basis. See the example on exact diagonalization if needed. In this case, we put this procedure into a function, with the option to scale the Coulomb interactions.
def diagonlize(scaleU=1):
umat = edrixs.get_umat_slater('d',
slater[0][1]*scaleU,
slater[1][1]*scaleU,
slater[2][1]*scaleU)
cfmat = edrixs.angular_momentum.cf_tetragonal_d(ten_dq=ten_dq, d1=d1, d3=d3)
H = edrixs.build_opers(2, cfmat, basis) + edrixs.build_opers(4, umat, basis)
e, v = scipy.linalg.eigh(H)
e = e - np.min(e) # define ground state as zero energy
return e, v
Let us look what happens when we run the function with the Coulomb interactions switched off and check the degeneracy of the output. Look at this python string formatting tutorial if the code is confusing.
e, v = diagonlize(scaleU=0)
e = e.round(decimals=6)
unique_e = np.unique(e)
degeneracies = [sum(evalue == e) for evalue in unique_e]
print("E \tDegeneracy")
for evalue, degenvalue in zip(unique_e, degeneracies):
print("{:.1f}\t{:.0f}".format(evalue, degenvalue))
print("{} distinct energies".format(len(unique_e)))
E Degeneracy
0.0 1
0.2 4
0.4 1
2.5 4
2.7 4
3.4 8
3.6 8
5.0 1
5.9 8
6.8 6
10 distinct energies
We see 10 distinct energies, which is the number of ways one can arrange two holes among 4 energy levels – which makes sense as the tetragonal field involves four levels \(zx/zy, xy, 3z^2-r^2, x^2-y^2\). To see what is going on in more detail, we can also calculate the expectation values of the occupancy number of the orbitals \(3z^2-r^2, zx, zy, x^2-y^2, xy\). To create the operator, first write the matrix in the real harmonics basis \(|3z^2-r^2,\uparrow>\), \(|3z^2-r^2,\downarrow>\), \(|zx,\uparrow>\), \(|zx,\downarrow>\), etc. In this basis, they take a simple form: only the diagonal terms have element 1. We therefore make a 3D empty array and assign the diagonal as 1. Check out the numpy indexing notes if needed.
Recalling the necessity to put everything in the same basis, we transform into the complex harmonic basis and then transform into our Fock basis
nd_complex_harmoic_basis = edrixs.cb_op(nd_real_harmoic_basis,
edrixs.tmat_r2c('d', True))
nd_op = edrixs.build_opers(2, nd_complex_harmoic_basis, basis)
We apply the operator and print out as follows. Check the numpy docs if the details of how the spin pairs have been added up is not immediately transparent.
nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op])
message = "{:>3s}" + "\t{:>6s}"*5
print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" ")))
message = "{:>3.1f}" + "\t{:>6.1f}"*5
for evalue, row in zip(e, nd_expt.T):
spin_pairs = row.reshape(-1, 2).sum(1)
print(message.format(evalue, *spin_pairs))
E 3z2-r2 zx zy x2-y2 xy
0.0 2.0 2.0 2.0 0.0 2.0
0.2 1.0 2.0 2.0 1.0 2.0
0.2 1.0 2.0 2.0 1.0 2.0
0.2 1.0 2.0 2.0 1.0 2.0
0.2 1.0 2.0 2.0 1.0 2.0
0.4 0.0 2.0 2.0 2.0 2.0
2.5 2.0 2.0 2.0 1.0 1.0
2.5 2.0 2.0 2.0 1.0 1.0
2.5 2.0 2.0 2.0 1.0 1.0
2.5 2.0 2.0 2.0 1.0 1.0
2.7 1.0 2.0 2.0 2.0 1.0
2.7 1.0 2.0 2.0 2.0 1.0
2.7 1.0 2.0 2.0 2.0 1.0
2.7 1.0 2.0 2.0 2.0 1.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.4 2.0 1.5 1.5 1.0 2.0
3.6 1.0 1.6 1.4 2.0 2.0
3.6 1.0 1.5 1.5 2.0 2.0
3.6 1.0 1.5 1.5 2.0 2.0
3.6 1.0 1.6 1.4 2.0 2.0
3.6 1.0 1.3 1.7 2.0 2.0
3.6 1.0 1.5 1.5 2.0 2.0
3.6 1.0 1.4 1.6 2.0 2.0
3.6 1.0 1.6 1.4 2.0 2.0
5.0 2.0 2.0 2.0 2.0 0.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
5.9 2.0 1.5 1.5 2.0 1.0
6.8 2.0 0.4 1.6 2.0 2.0
6.8 2.0 1.3 0.7 2.0 2.0
6.8 2.0 1.0 1.0 2.0 2.0
6.8 2.0 1.0 1.0 2.0 2.0
6.8 2.0 1.0 1.0 2.0 2.0
6.8 2.0 1.2 0.8 2.0 2.0
The lowest energy state involves putting both holes in the \(x^2-y^2\) orbital, which makes sense. Now, let us redo the proceedure including Coulomb repulsion, which imposes an energy cost to putting multiple electrons in the same orbital.
e, v = diagonlize(scaleU=1)
nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op])
message = "{:>3s}" + "\t{:>6s}"*5
print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" ")))
message = "{:>3.1f}" + "\t{:>6.1f}"*5
for evalue, row in zip(e, nd_expt.T):
spin_pairs = row.reshape(-1, 2).sum(1)
print(message.format(evalue, *spin_pairs))
E 3z2-r2 zx zy x2-y2 xy
0.0 1.0 2.0 2.0 1.0 2.0
0.0 1.0 2.0 2.0 1.0 2.0
0.0 1.0 2.0 2.0 1.0 2.0
2.4 1.2 2.0 2.0 0.8 2.0
2.5 1.0 2.0 2.0 1.0 2.0
2.5 1.0 2.0 2.0 2.0 1.0
2.5 1.0 2.0 2.0 2.0 1.0
2.5 1.0 2.0 2.0 2.0 1.0
3.2 1.8 1.2 1.8 1.2 2.0
3.2 1.8 1.7 1.3 1.2 2.0
3.2 1.8 1.5 1.5 1.2 2.0
3.2 1.8 1.7 1.3 1.2 2.0
3.2 1.8 1.4 1.6 1.2 2.0
3.2 1.8 1.5 1.5 1.2 2.0
4.0 2.0 1.9 1.9 1.1 1.1
4.0 2.0 1.9 1.9 1.1 1.1
4.0 2.0 1.9 1.9 1.1 1.1
4.3 0.9 2.0 2.0 1.2 1.9
4.7 1.4 1.4 1.6 1.9 1.7
4.7 1.4 1.5 1.5 1.9 1.7
4.7 1.4 1.6 1.4 1.9 1.7
4.7 1.4 1.4 1.6 1.9 1.7
4.7 1.4 1.4 1.6 1.9 1.7
4.7 1.4 1.6 1.4 1.9 1.7
4.9 1.0 2.0 2.0 2.0 1.0
5.5 2.0 2.0 2.0 1.0 1.0
5.6 1.8 1.6 1.4 1.2 2.0
5.6 1.8 1.4 1.6 1.2 2.0
6.5 1.2 1.6 1.4 1.8 2.0
6.5 1.2 1.4 1.6 1.8 2.0
6.8 1.8 1.4 1.6 1.9 1.3
6.8 1.8 1.5 1.5 1.9 1.3
6.8 1.8 1.5 1.5 1.9 1.3
6.8 1.8 1.6 1.4 1.9 1.3
6.8 1.8 1.4 1.6 1.9 1.3
6.8 1.8 1.6 1.4 1.9 1.3
7.4 2.0 1.1 1.1 1.9 1.9
7.4 2.0 1.1 1.1 1.9 1.9
7.4 2.0 1.1 1.1 1.9 1.9
8.0 2.0 1.8 1.8 2.0 0.4
8.5 2.0 1.5 1.5 2.0 1.0
8.5 2.0 1.5 1.5 2.0 1.0
9.3 2.0 1.0 1.0 2.0 2.0
9.4 2.0 1.0 1.0 2.0 2.0
12.8 1.9 1.2 1.2 1.9 1.7
Now the lowest energy state involves splitting the holes between the two highest energy \(x^2-y^2\) and \(3z^2-r^2\) orbitals. i.e. we have gone from low-spin to high spin. Working out the balance between these two states is often one of the first things one wants to determine upon the discovery of an interesting new material [1].
Footnotes
Total running time of the script: (0 minutes 0.199 seconds)